Monte Carlo Simulation

Clewlow and Strickland Chapter 4
DATA 5695/6695

Tyler J. Brough

2025-02-25

Clewlow and Strickland: Chapter 4 Monte Carlo Simulation

Valuation by Simulation


  • We have previously shown that the value of an option is the risk-neutral expectation of its discounted payoff

  • In the binomial model this was seen in the single-period model

\[ C_{0} = e^{-rh}\left[p^{\ast} C_{u} + (1 - p^{\ast})C_{d}\right] \]

  • The term inside the brackets is the binomial expected value under the risk-neutral distribution

  • This suggests a path forward for models beyond the binomial model

  • We can obtain an estimate of the risk-neutral expected value by computing the average of a large number of discounted payoffs

  • This is a LLN argument based on frequentist reasoning

Valuation by Simulation (cont’d)

  • Consider an European call option that pays \(C_{T}\) at expiry.

  • First, we simulate the risk-neutral process of the state variables from their observed values today

  • Second, apply the payoff function to the terminal values simulated along each path and obtain \(C_{T,j}\) for path \(j\)

  • Then discount this payoff (assuming a constant risk-free rate):

\[ C_{0,j} = e^{-rT} C_{T,j} \]

  • Repeat this \(M\) times and obtain:

\[ \hat{C}_{0} = \frac{1}{M} \sum\limits_{j=1}^{M} C_{0,j} \]

Valuation by Simulation (cont’d)


  • This is basically a proposal to evaluate the expectation statistically using Monte Carlo integration

  • There will thus be sampling error introduced in the algorithm and we can estimate it via the standard error of the simulation

\[ SE(\hat{C}_{0}) = \frac{SD(C_{0,j})}{\sqrt{M}} \]

The standard deviation is given by:

\[ \mathrm{SD}\left(C_{0,j}\right) = \left(\frac{1}{M-1} \sum_{j=1}^M\left(C_{0,j} - \hat{C}_0\right)^2\right)^{\frac{1}{2}} \]